The introduction in Italy of the North American lagomorph Sylvilagus floridanus, simply named cottontail, has affected the predator-prey dynamics of native European hares and foxes, by increasing fox's predation rate on hares. This results in a sort of apparent competition, for which the presence of a new prey increases predation on another native prey, and could ultimately negatively impact hares. In addition, cottontails could carry several potentially harmful viruses and parasites, especially the European brown hare syndrome (EBHS) virus, which has devastating effects on hare populations. The cottontails' role in the disease spreading is still not known. The model introduced here for possible disease containment assumes that all foxes are healthy carriers and that they could infect other animals through the environment. The analysis allows the characterization of the system's equilibria. The rather exhaustive investigation of the possible system transitions among stable configurations would help the applied ecologist in the field work